3 Answers
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Chronology is certainly one really good way to organize the books of the Bible, and some people may prefer this. There certainly are editions of the Bible in chronological order that are available. It should be noted that the order of the books of the Bible is not inspired, so you are really free to organize them however you want.

One of the problems with chronology, though, is the overlapping of some accounts. The book of Acts covers quite a bit of history, so the epistles occur throughout it. Furthermore, there are some books that are not historical in nature, like the Psalms, Proverbs and Ecclesiastes. Sure, they were written mostly by David and Solomon, but there really isn't a lot of reason to place them in between chapters of 1 and 2 Kings.

That being said, there is a common order that does have a purpose. This historical narratives in the Old Testament are pretty much in chronological order. The prophets span several kings and are not written historical narrative but has prophetic utterances in history.

History Narrative (chronological, for the most part, with some overlapping)

Wisdom Literature

Major Prophets

Minor Prophets

Gospels (generally chronological, but each overlaps the others)

History (Acts) - chronological

Letters to Churches, named for cities

Letters to Individuals, named for recipient

Letters by People, named for authors

Prophecy

So, again, order them however you want for your personal reading. If you don't like the common order, that's totally fine. I remember an old mentor of mine reading through the Psalms backwards, which was an interesting idea.

Part of the problem is that some books cover overlapping time periods. How do you order the Gospels for example. John for example even hits on creation briefly. This would complicate things greatly since there isn't any way to logically order them.

In many cases, things are Chronologically ordered where it makes sense. Genesis comes at the beginning, Revelation comes at the end, New Testament comes after Old.

Another problem is how do you judge prophecy or things like psalms that don't deal with a particular time period or deal with a much later time period? There are lots of issues in determining just what chronological order would make sense.

You could try doing Chronological order by writing, but do you then do it by the first time it is written down or the time that it was original presented? Again, some points can get tricky as some were assembled over generations which can overlap.

Thus we end up with the common approach being a loose chronological order that is subgrouped by category to help resolve overlapping books that cover different purposes.

In which Bible, there are plenty of translations, and the order depends on who was the translator/editor, since you can order the Bible books as you wish.

"Don't we have a good idea of when each was written? Wouldn't it make more sense?"

We know partially who write what. For example, for centuries we thought the Hebrews epistle was authored by Paul, but it was not as many "recent" studies demonstrate.
Ordering by author would make a result very much like ordering chronologically, since most of books are from different centuries.